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Macroscopic control of high-order harmonics

quantum-path components for the generation of

attosecond pulses

H. Merdji, M. Kovačev, W. Boutu, P. Salieres, F. Vernay, B. Carré

To cite this version:

H. Merdji, M. Kovačev, W. Boutu, P. Salieres, F. Vernay, et al.. Macroscopic control of high-order

harmonics quantum-path components for the generation of attosecond pulses. Physical Review A,

American Physical Society, 2006, 74 (4), �10.1103/PhysRevA.74.043804�. �hal-02145556�

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Macroscopic control of high-order harmonics quantum-path components

for the generation of attosecond pulses

H. Merdji,1M. Kovačev,2 W. Boutu,1 P. Salières,1F. Vernay,1 and B. Carré1

1CEA/DSM/DRECAM/Service des Photons, Atomes et Molécules, Bât. 522, CEA-Saclay, 91191 Gif-sur-Yvette, France 2Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

共Received 22 March 2006; published 4 October 2006兲

We present measurements of the different quantum path contributions to the high-order harmonic emission. Through spatial and spectral filtering, we evidence the strong correlation between the spatial and spectral distributions, which allows us to quantify the contribution of each quantum path. A systematic analysis as a function of the generating parameters has been done to identify the conditions for efficient generation and selection of a single quantum path. We show that combining phase matching and spatial filtering allows maximizing and selecting the short quantum path contribution, condition for the generation of “clean” and intense attosecond pulses.

DOI:10.1103/PhysRevA.74.043804 PACS number共s兲: 42.65.Ky, 42.65.Re, 42.87.Bg

I. INTRODUCTION

The generation of the high-order harmonics共HHG兲 of in-tense laser pulses focused in gases is attracting much atten-tion due to both its fundamental and applied interest关1,2兴.

Indeed, the advanced characterization of the process both gives insight into the ultrafast atomic dynamics in the laser field, and qualify this source of XUV radiation for potential applications. In particular, measurements of the spatial关3–7兴

and temporal 关8–11兴 coherence properties have allowed a

deeper understanding of the generation process, and have revealed their good quality, which is unique in this spectral range. Together with the short pulse duration and the high energy 共up to microjoule energy per pulse 关12,13兴兲, these

properties have found applications in pump-probe experi-ments, e.g., for interferometry in plasma physics关14,15兴, but

also in solid state physics and in atomic and molecular spec-troscopy 关16兴. The good coherence also makes the HHG

source particularly suitable for seeding soft-x-ray lasers关17兴

or free-electron lasers 关18兴. Finally, the spectral coherence

over the extended harmonic spectrum is the key for the gen-eration of attosecond pulses: the high harmonic source has opened the field of the so-called attoscience which has been spectacularly growing in the last few years共for a review, see Ref.关19兴兲.

The major above breakthroughs rely on the deep theoret-ical understanding and experimental mastery of HHG. Pro-found insight in the generation process has been provided by the semiclassical three-step model关20,21兴, in which an

elec-tron first tunnels out of the atomic potential, is then acceler-ated in the laser field and finally driven back to the ion lead-ing to recombination on the ground state with emission of a burst of XUV light. Quantum description of HHG within SFA关22兴 and TDSE 关23–25兴, as well as direct experimental

probe of the electron dynamics关26兴, have grounded the

no-tion of electron trajectories in the laser field, the so-called quantum paths. For harmonic of qth order in the plateau, mainly two quantum paths contribute to the radiating dipole

Dq. The two paths differ by the ionization time tiand

emis-sion time te, or equivalently the travel time␶= te− ti of the

electron wave packet in the continuum. For the so-called

“short” trajectory 共index j=1兲, the travel time1 is of the order of half the optical period, whereas for the “long” tra-jectory共j=2兲,␶2is close to the optical period. The dipole Dq

共equivalently the local nonlinear polarization兲 can thus be expressed as the sum Dq= D1ei␾1+ D2ei␾2 of the two contri-butions. In the Lewenstein model关22兴, the phase␾j

identi-fies to the quasiclassical action along the j path and therefore depends on the parameters tej,␶j, laser intensity I and XUV

frequency q共the order q is considered as a continuous vari-able denoting the XUV frequency in␻unit兲. Note that, ulti-mately, the times tej,␶jare in turn functions of I and q, so

that for each class of trajectories the phase␾jis completely

determined by the laser intensity and the harmonic order q. One further demonstrates that the phase differential can be written as dj=␣jdI + tejdq, where the partial derivativejis

determined by the travel time␶j and is therefore larger for

the long trajectories than for the short ones. The ␣ and te

coefficients vary slowly with the frequency q. In the cutoff, the two classes of trajectories merge into one.

After propagation, the macroscopic XUV field in the pla-teau is the sum of two terms, refered to as ␶1- and

␶2-contributions, respectively,

Eq= E1ei⌽1+ E2ei⌽2. 共1兲 If we assume conditions close to phase-matching, the mac-roscopic phases take the simple form ⌽j⬇qL+␾j, where

L is the laser phase including the −␻t term. On the one

hand, the q dependence of the phases in Eq.共1兲 has

impor-tant consequences for the generation of attosecond pulses when they are obtained as a coherent sum of quasi-phase-locked共discrete harmonic or continuous兲 components Eq in

the plateau 关26–28兴. On the other hand, the I dependence,

i.e., the temporal and spatial variations, of the phases⌽j共␾j

determines the coherence properties of Eq, and in particular

its spectral and spatial characteristics关1兴. First, it is

respon-sible for the intrinsic chirp of the harmonic emission. Sec-ond, it also conditions the spatial phase and therefore the spatial profile of the XUV field. The different I dependence of the two quantum paths components 共different ␣j兲 thus

leads to a spatial and spectral separation of their

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tions to the macroscopic harmonic field关24,29兴. The

separa-tion has been used to study their different temporal coher-ence 关8,9兴 and phase matching 关10兴. However, to our

knowledge, there has been no experimental study of the am-plitudes E1 and E2, i.e., of the relative weights of the two contributions.

The discrimination and control of the␶1and␶2 contribu-tions are of great importance, first for a fundamental insight on HHG and second for controlling the XUV emission. Theoretically, the relative weight of the ␶1- and

␶2-contributions to the single atom response共i.e., D1/ D2兲 is still somewhat controversial: the SFA and TDSE approaches lead to different predictions, the ␶2 contribution being usu-ally larger in SFA共and presumably overestimated兲 than in TDSE 关24,25兴. Experimental studies are thus needed. Note

that the relative weight of the macroscopic contributions 共E1/ E2兲 is also affected by the propagation, i.e., by the way phase matching is achieved for each contribution. On the control side, producing an XUV field Eq at given frequency,

with definite spectral and spatial characteristics, implies in general that only one Ejcontribution is selected in Eq. 共1兲,

thus limiting the space and time phase variation. Moreover, the production of attosecond pulses from emission in the plateau demands to select one single quantum-path, the con-dition for synchronized共phase-locked兲 emission over a broad spectral range关30–34兴. In the case where the two classes of

trajectories contribute, two bursts are emitted with different timings every half cycle, which blurs the attosecond struc-ture. In a number of recent works 关26,27,32,33,35,36兴 the

subcycle pulses in HHG have been characterized for a set of harmonics—up to 30 harmonics in Ref. 关26兴—from

mea-surement of the harmonic relative phases共dj= tejdq at fixed I兲. They have revealed the possibility to generate pulses as

short as 130 as that could be further compressed after com-pensation of their intrinsic chirp关26,33,37兴. The experiments

assume and partially confirm that, for appropriate generation conditions, propagation and spatial filtering selects mainly the contribution of one quantum path 共namely, the “short” path兲. Actually, more accurate characterization of the sub-cycle electron dynamics and even shorter attosecond pulses could be achieved from a direct monitoring and a more complete discrimination of the paths contributions.

In this paper, we present measurements of the contribu-tions of the two quantum paths. The control/discrimination of the␶1and␶2contributions can be envisaged at two levels. At the “upstream” level, one favors one particular contribu-tion by choosing the appropriate condicontribu-tions in the generacontribu-tion process itself. This is made possible by adjusting the focus-sing geometry of the laser beam, then ufocus-sing propagation and phase-matching as a “filter.” At the “downstream” level, one takes advantage of the different spectral and spatial charac-teristics of the E1 and E2 components to discriminate them through spectral or spatial filtering. We demonstrate the pos-sibility of combining upstream and downstream control of the ␶1 and ␶2 contributions. In the downstream control, we show in turn how the filtering in the spatial domain can be consistently combined with the one in the spectral domain, and the relative weight of the␶1 and␶2 contributions esti-mated. Finally, we study the variation of this relative weight as a function of different generation parameters.

II. EXPERIMENT

A. Identification of path contributions

We first explain the principle of the upstream and down-stream controls of the␶1 and␶2components. As mentioned, it is based on the different phase properties of the␶1 and␶2 contributions to the atomic dipole for the upstream control, to the related macroscopic field for the downstream control. Upstream control involves the spatial characteristic of the dipole phase ␾j. In the medium, the macroscopic field Eq builds up along propagation under the phase-matching condition关38兴

ⵜជ⌽j⬇ ⵜជ共qL+␾j兲, 共2兲

i.e., equality of the wave vectors of the harmonic field and nonlinear polarization, where ⵜជ ␾j=␣jⵜជI. In the case of a

free propagating laser beam focused in a gas jet, it has been demonstrated that the smaller gradient ␣1ⵜជI for the short trajectory in Eq.共2兲 共␣1⬍␣2兲 determines its dominant con-tribution to on-axis emission, when the laser beam is focused before the jet关24,29,38兴. Conversely, the large gradient␣2ⵜជI

for the long trajectory determines its dominant contribution to off-axis emission, when the laser beam is focused after the jet. The two contributions are therefore discriminated up-stream through the focussing geometry. When the two con-tributions are significantly present after propagation, they are also separated downstream in the spatial domain, in the form of an inner and an outer region in the far-field profile.

Together with the spatial discrimination, the two contri-butions are discriminated downstream in the spectral do-main. Hence in a real laser pulse where I varies in time 共within adiabatic approximation for HHG兲, the intensity-dependent term in the phase ⌽j 共␾j兲 causes a frequency

modulation, a chirp, of the harmonic emission关29兴:

⌬␻共t兲 = −⳵⌽j共t兲

t = −␣j

I共t兲

t . 共3兲

It is clear from Eq.共3兲 that the spectral width ⌬1

关propor-tional to 共⳵⌽1/⳵t兲max兴 of the E1 contribution in Eq. 共1兲 will be smaller than that of E2. Thus, spectral or spatial filtering of the macroscopic harmonic field should allow a downstream selection of either␶1or␶2 contribution.

In a first experiment, we perform a similar analysis as in Refs.关8,9兴 in order to identify the two contributions

accord-ing to the above discrimination in the spectral and spatial domains. The experiment was carried out at the femtosecond LUCA laser facility of CEA/DRECAM in Saclay 共titanium-:sapphire system at 800 nm, 80 mJ, 60 fs, 20 Hz兲. The laser energy is adjusted using a diaphragm. Two phase-locked IR pulses of ⬃1 mJ energy are produced in a Michelson type interferometer and focused with a 1 m focal length lens 4 mm after an argon jet共1 mm length兲 at a backing pressure of 900 Torr. The interferometer is identical to the one used in Refs.关39,40兴. It allows for generating two phase-locked

har-monic pulses separated in space and delayed in time, which allows for delay-dependent interference in the far-field. As shown in Fig.1, XUV light is analyzed with a monochro-mator consisting in a toroidal mirror, a grating, and an exit

MERDJI et al. PHYSICAL REVIEW A 74, 043804共2006兲

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slit. The width of the slit can be adjusted to tune the spectral acceptance from few nanometers共selection of one harmonic order with full spectral profile transmitted兲 to 0.05 nm 共har-monic spectral analysis兲. The far-field spatial distribution is measured on microchannel plates coupled to a phosphor screen and a 12 bit-CD camera. Under assumption of good mutual coherence of the two harmonic sources, the temporal coherence can be mapped in the beam cross section from the far field interference pattern, by varying the delay between the two pulses共laser pulses delay in the Michelson兲. The slit in the spectral plane was adjusted to select a single harmonic order. We will consider in this study the 17th harmonic which is representative of the behavior in the plateau region. In the inset in Fig. 2, the far-field interference pattern for 17th harmonic, measured in single shot and at zero delay between the two pulses, clearly reveals the two regions that we associate to the contributions of the short共inner re-gion for on-axis emission兲 and long trajectory 共outer rere-gion for off-axis emission兲. The fringes visibility defined as 共Imax− Imin兲/共Imax+ Imin兲 is plotted in Fig. 2 as a function of

the delay, for the inner and outer regions. Because of the highly controlled, coherent physics of the harmonic genera-tion, the two mutually coherent harmonic sources should produce a fringe visibility close to 1: the two sources are said to be “phase locked.” In practice, slightly different condi-tions in producing the two sources lead to a visibility be-tween 0.4共long trajectory兲 and 0.9 共short trajectory兲 at zero delay. Surprisingly we measure a lower fringe visibility in the region associated to the long trajectory in contrast to previous results关8兴 where similar 共visibility around 0.5兲

val-ues were obtained for each contribution. The different phase locking that we measured for the short and the long trajec-tories may be due to phase matching effects. In particular, the electronic and atomic dispersions in the two focal regions are locally different and this may play a role for on- and off-axis field construction关10兴. Consequently the two-source

phase locking might be affected differently for each trajec-tory. However, we should note that this does not affect the following analysis. The coherence time, i.e., the inverse of the spectral width ⌬q, is defined as the full width at

half maximum 共FWHM兲 of the visibility in Fig. 2. The field in the inner region has a long coherence time 共24 fs, ⌬1⯝0.3±0.05 nm兲, whereas it has a shorter one 共8 fs, ⌬2⯝0.9±0.05 nm兲 in the outer region. This confirms the identification of the inner field to the ␶1 contribution 共small spectral width兲 and of the outer field to the␶2 contri-bution共large spectral width兲. Our conclusions and measured coherence times are in good agreement with Refs. 关8,9兴.

Considering that the spectral width mainly reflects the intrin-sic chirp关see Eq. 共3兲兴, we get an estimate of the␣2/␣1ratio of the phase coefficients from the coherence times; we find

␣2/␣1⯝3. Although this parameter may not directly scale with␣2/␣1due to the influence of phase matching, we note that the divergence of the field in the outer region is about 3 times larger than in the inner one关24兴.

B. Spectral and spatial filtering

We can further evidence the spectral/spatial correlation by introducing a “downstream” filtering of the HHG generated using only one arm of the Michelson interferometer. To this purpose, a motorized adjustable diaphragm is installed in the far-field just before the monochromator. The method is two-fold: 共i兲 we filter out the outer region in the spatial profile with the HHG diaphragm, and monitor the spectral profile, and共ii兲 we filter out the wings of the spectral profile with the variable slit and monitor the spatial profile in the far field. The laser beam is focused 4 mm after the argon gas jet in order to maximize the long trajectory contribution. The laser energy is varied around 1 mJ共intensity ⯝5⫻1013W cm−2at focus兲 in order to minimize self phase modulation of the fundamental pulse via ionization of the gas medium that would induce a blueshift of the laser and subsequently of the harmonic spectra.

Spatial filtering. In the first filtering operation共i兲, we

con-sider the spatial profile of H17 in Fig.3共a兲. It clearly exhibits the inner and outer regions of the inset in Fig.2, associated to ␶1 and ␶2 contributions, respectively. Now, we vary the HHG diaphragm from 10 mm共full beam兲 to 1 mm. The

sig-Gas jet DiaphragmA djustable

A djustable Slit

Far field detection: MCP +CCD camera Grating Spectral detection: movable photomultiplier T oroidal mirror IR laser HHG

FIG. 1. Experimental setup for the spectral and spatial analysis of quantum path contributions.

-20 -15 -10 -5 0 5 10 15 20

0 1

Fring evisibility

Delay (fs)

FIG. 2. 共Color online兲 Average visibility of the fringes in the far-field interference pattern for the 17th harmonic, generated in argon, as a function of delay共1st-order autocorrelation trace兲: short 共full circle兲 and long 共open triangle兲 trajectories contributions. In dotted line are shown the Gaussian fits of the measured traces. The FWHM gives the coherence time. Inset: Interference pattern re-corded at 0 delay.

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nal decreases and, more importantly, is spectrally narrowed as shown in Fig.3共b兲. We relate this narrowing to a substan-tial cut of the␶2contribution operated in the spatial domain. The evolution of the integrated harmonic signal and of the spectral width 共FWHM兲 as a function of the HHG beam aperture is shown in Fig.4, for two laser energies 0.85 and 1.25 mJ with a laser diaphragm of 12.5 mm. The larger widths at 1.25 mJ reflect the intensity-dependent broadening, ⌬j⬃␣j共⳵I /t兲 due to the intrinsic chirp 关10兴. For both

ener-gies, the maximum values, 0.5 and 0.6 nm for full HHG beam aperture, correspond to the profile under which the spectrally narrow␶1contribution共⬃0.3 nm兲 is superimposed to the dominant and spectrally large ␶2 contribution 共⬃0.9 nm兲; the resulting effective FWHM is therefore in be-tween those of the two fields. The minimum value, around 0.35 nm for both energies, is comparable to that of ␶1 contribution.

In Fig.4共a兲, the integrated signal is compared to the one derived from the spatial profile in Fig.3共a兲, simulating nu-merically the transmission of the HHG diaphragm. In the

latter, we have parametrized the spatial profiles associated to the␶1共central structure approximated as a Gaussian profile兲 and ␶2 contributions 共pedestal approximated as a super Gaussian profile兲. The intensity of each trajectory contribu-tion corresponds to the integrated signal under each curve simulating the␶1 and␶2profiles. The long trajectory contri-bution is 1.6 times larger than the short trajectory contribu-tion for the full beam. Then, we can estimate the energies

E1共d兲 and E2共d兲, respectively, in the1 and␶2 contributions by taking into account their respective divergence extracted from Fig. 3共a兲. The total energy E1共d兲+E2共d兲 transmitted through the HHG diaphragm of diameter d plotted in Fig.

4共a兲compares satisfactorily with the integrated spectral pro-file; the plot of E1共d兲 and E2共d兲 illustrates how the HHG diaphragm changes the relative weight of the two contribu-tions. Now, we want to correlate, at least semiquantitatively, the variation of the␶1 and␶2contributions to the one of the spectral widths in Fig.4共b兲. For this, we assume that for each contribution, the spectral profile can be represented by a Gaussian function Gj共␭兲, with width ⌬j. We can retrieve the

-4 -2 0 0 50 100 150 200 τ1 τ2 Diaphragm (a) Intensity (arb.units) Radialdistance (mm) 2 4 -2 -1 0 10-4 10-3 10-2 10-1 (b) opendiaph closeddiaph Intensity (arb.units) ∆λ(nm) 1

FIG. 3. 共a兲 Spatial profile of the 17th harmonic generated in argon with a laser energy of 0.85 mJ when the laser is focused 4 mm after the gas jet. The laser diaphragm is 12.5 mm. Fits of␶1 共Gaussian, dotted line兲 and␶2contributions共superGaussian, dashed line兲. The arrow shows the size of the 1 mm closed HHG dia-phragm. 共b兲 Corresponding spectral profiles for an open and a closed HHG diaphragm. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 4 10 0.3 0.4 0.5 0.6 (b) Spectral FWHM (nm) Diaphragm diameter (mm) 1.25mJ 0.85mJ Simulat 1.25mJ Simulat 0.85mJ (a) Normalized Signal Expt. Simul τ1+τ2 τ1 τ2 2 6 8

FIG. 4. 共a兲 Integrated spectral intensity of H17 generated in argon as a function of the HHG diaphragm diameter, compared to the simulated signal, sum of the ␶1 and ␶2 contributions. The 12.5 mm apertured, 0.85-mJ laser pulse is focused 4 mm after the gas jet.共b兲 HHG spectral width for two laser energies; the experi-mental widths are compared to the effective Gaussian width of the simulated profiles.

MERDJI et al. PHYSICAL REVIEW A 74, 043804共2006兲

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spectral profile of the light filtered, by forming the quantity

E1共d兲 ⌬1 G1共␭兲+

E2共d兲

⌬2 G2共␭兲, in which the Gj共␭兲 are pondered by the spectral densities Ej共d兲

j in each of the␶j contributions. In

the simulation, the widths⌬iare chosen close to their

esti-mated values in Fig.2. The effective width共FWHM兲 of the simulated profile is compared to the measured width at 0.85 mJ in Fig. 4共b兲. Similar agreement is obtained for the spectral profile modelization at 1.25 mJ laser energy. The agreement between measured and simulated widths should be considered as semiquantitative. It evidences the clear cor-relation between the variation of the spatial and spectral pro-files. It allows to assign this correlation to the differential filtering, in the spatial domain, of the␶1and␶2contributions. By filtering the outer region of the spatial distribution, we obviously do not “cut” the full␶2 component, but reduce its contribution below 10% of the total signal.

Spectral filtering. Conversely, in the second filtering

op-eration共ii兲 on H17, using a variable slit in the monochro-mator, we “cut” the spectral distribution and monitor the spa-tial one. Figure5displays the spatial profiles of H17 for two slit sizes, respectively transmitting the full spectrum共opened slit兲 and selecting the central width ⌬␭=0.2 nm. For the closed slit, the outer region of the spatial profile is almost completely suppressed, the central structure is narrower.

Symmetrically to filtering共i兲, we can clearly correlate the wings of the spectral profile to the outer region of the spatial distribution, related to the same field, i.e., the␶2contribution. In Fig.5, the reduction of the integrated signal is not as large but still comparable with the one measured in Fig.4共a兲for a small diaphragm aperture.

Finally, the filtering operations 共i兲 and 共ii兲 show that we can, in a consistent way, estimate the weight of the␶1and␶2 contributions from either the spatial profile of the far field, or the spectral profile. The filtering operation共i兲 in the spatial domain appears even easier and more efficient than in the

spectral one. The experimental evidence and our simple analysis are at least in qualitative agreement with the full simulations by Gaarde et al. 关24兴, where the atomic dipole

was calculated from time-dependent Schrödinger equation and propagation fully taken into account.

C. Variations of the path contributions

We illustrate now how we can analyze the path contribu-tions when some of the generation parameters are varied. First, we have studied the spectral and spatial profiles of the harmonic emission as a function of the jet/focus position 共z=zfocus− zjet⬎0 for focus after the jet兲. They are, respec-tively, shown for H17 in Figs. 6共a兲 and 6共b兲 共laser energy

E = 1.25 mJ, laser diaphragm d = 12.5 mm兲. As observed in

the previous studies either in the spatial关9,41兴 or in the

spec-tral关10,42兴 domain, we measure a strong dependence of the

profiles with the focus position. Our measurement of both quantities allows us to correlate their variation. When the laser is focused before the gas jet共z=−4 mm兲, the spectral and spatial profiles are narrow. When the focus moves into 共z=0 mm兲 and after 共z= +4 mm兲 the gas jet, the total signal increases, the spectral profile broadens and a pedestal

ap--2 -1 0 10-4 10-3 10-2 10-1 100 -3 -2 -1 0 2 0 50 100 150 Intensity (arb.units) Radial distance (mm) openslit closedslit slit δλ (nm) 1 3 1

FIG. 5. Spatial profiles of the 17th harmonic generated in argon when the slit in the spectral plane is opened or closed to ⌬␭=0.2 nm. The 1-mJ laser pulse is focused 4 mm after the gas jet. The inset shows the corresponding spectral profile.

-4 0 4 0 60 120 180 (b) Intensity (arb.units) Radial distance (mm) 46 47 48 10-4 10-3 10-2 10-1 (a) Intensity (arb.units.) Wavelength (nm) z = -4mm z = 0mm z = +4mm z = -4mm z = 0mm z = +4mm

FIG. 6. Measured spectral共a兲 and spatial 共b兲 profiles of the 17th harmonic as a function of the gas jet position relative to the focus. The laser energy is 1.25 mJ and the laser diaphragm is 12.5 mm.

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pears in the spatial distribution. We attribute the broadening of both the spectral and spatial profiles to the onset of E2 associated with the␶2contribution, whereas the central struc-ture is mainly attributed to E1 associated with ␶1. At large

z⬎ +4 mm, the signal decreases; the profiles are narrow

again. The variations of the spectral and spatial profiles are clearly correlated as a function of focusing. First, they reflect how z-dependent phase matching can favor the one or the other contribution: this is the principle of the up-stream con-trol of HHG. Second, for phase-matched emission, they re-flect the dependence of the field parameters 共chirp兲 on the laser intensity 关10兴. A more quantitative analysis is

per-formed in Fig.7共a兲, where, together with the integrated sig-nal, we have plotted the relative weight of␶1contribution共in percentage of total signal兲 as a function of the jet to focus position z. The␶1 weight is obtained from a fit of the 1D radial profile in Fig. 6共b兲 as a sum of Gaussian/

superGaussian components关see also Fig. 3共a兲兴, and further 2D integration. The␶1 contribution is slightly higher when the laser is focused before the gas jet共between z=−10 mm and z = 0 mm兲. For z⬎0, the ␶2 contribution increases sig-nificantly up to 60% of the total signal. This corresponds to a phase matching achieved in the off-axis region of the gas jet, involving a large radial gradient␣jⵜជ共I兲 of the laser intensity

in Eq. 共2兲, and resulting in an off-axis XUV emission as

shown in the spatial profiles. For z⬎ +10 mm, the phase matching is again realized preferentially on axis for the ␶1 contribution. We find the same generic behavior using a larger diaphragm and a lower laser energy关triangles in Fig.

7共a兲兴. We naturally observe that with smaller confocal pa-rameter and intensity, the harmonic signal varies more rap-idly with z. Note that the signal dependence indicates that when ␶2 component is phase matched, the total signal is maximum.

In Fig.7共b兲, we have plotted the weight of the ␶1 contri-bution for H17 as a function of the laser energy, for

z = + 2 mm and diaphragm d = 12.5 mm. The␶1 contribution slightly dominates at low energy and then decreases when energy increases. When the harmonic total output is opti-mized 共E=1.25 mJ兲, ␶2 becomes dominant. Qualitatively, this variation reflects the intensity-dependent phase matching of ␶1 and ␶2 contributions: the ␶2 component is efficiently phase matched for large radial gradient obtained at high in-tensity. The further increase of␶1contribution may be attrib-uted to the relative degradation of phase matching in the presence of strong ionization and electronic dispersion, which is more critical off axis共large gradients兲 for␶2than it is on axis for␶1. The total signal consequently decreases.

D. “Up-” and “down-stream” filtering of one contribution

Our analysis shows how control and filtering of the path contributions can be combined, provided that the generation conditions are properly defined. First we can control the quantum path contributions with the energy of the generating laser. Hence, as previously shown in Fig.7共b兲, one may fa-vor either the short or the long trajectory by slightly tuning laser energy. Second, we have an additionnal control of quantum path contributions with the focusing geometry. To optimize the␶1contribution in our geometry, there is advan-tage to focus the laser close to the gas jet at z⯝0:␶1 con-tributes to 50% of the total signal which is close to its maxi-mum. Additionnaly we have put a diaphragm to suppress the off-axis signal and we found a␶1contribution that is around one order of magnitude larger than that of␶2. The measured

E1/ E2 ratio for the full HHG beam and for the beam aper-tured at 1 mm at various focus positions and at two laser energies is reported in Table I. By putting a␾= 1 mm dia-phragm in the far field, we measure a contribution from the short quantum path that strongly dominates. The above fil-tering can be used for any harmonic in the plateau region. As a result, almost “pure”␶1contribution for a set of harmonic components will be obtained. The long trajectory can also be selected. From Fig.7共b兲, we note that ␶2 emission is maxi-mum共60% of the total signal兲 at z= +4 mm, where the

har-0.3 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.4 0.5 0.6 0.70.7 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (b) Laser energy (mJ) -8 -6 -4 -2 0 2 4 8 10 12 14 -12 -10 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (a) jet/focus positionz(mm)

Harmonic signal (arb.units)

Normalized τ1 contribution Normalized τ1 contribution

Harmonic signal (arb.units)

FIG. 7. 共a兲 Contribution of the short trajectory for H17 as a function of the relative jet-to-focus position z at E = 1.25 mJ and laser diaphragm d = 12.5 mm共open circle兲. The same dependance together with the total harmonic signal are shown for a laser energy of 0.85 mJ and d = 14 mm共respectively, open and full triangles兲. 共b兲 Contribution of the short trajectory as a function of the laser energy at z = + 2 mm and d = 12.5 mm. The total harmonic signal is plotted in full circle. The error bars共not shown兲 do not exceed 10% of the signal.

MERDJI et al. PHYSICAL REVIEW A 74, 043804共2006兲

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monic signal is high. We can select␶2just by putting a disk inside the beam to stop the on-axis emission where ␶1 is emitted关43兴.

This analysis is crucial for the generation of clean and intense attosecond pulses. The cutoff region of the harmonic spectrum, where a single path contribute to the emission, has been used to produce isolated attosecond pulses关28,44兴. In

order to produce shorter and more intense pulses, the plateau region should be used, and this applies both to trains and to isolated pulses obtained through polarization gating 关34,45–47兴. But the contribution of the two quantum paths,

with a relative weight determined both by the atomic dipole and the z-dependent phase matching, directly affects the phase locking between consecutive harmonics 关30–32兴 and

blurs the attosecond structure. The generation of a single attosecond burst per half optical cycle thus requires that a single quantum path is selected.

III. CONCLUSION

In summary, our measurements demonstrate that we can now have a complete control of the quantum path contributions in high harmonic generation. The two contri-butions, ␶1 and ␶2 for, respectively, the short and the long quantum paths, are identified from their strongly correlated properties in the spatial and spectral domains. We have stud-ied them as a function of generation parameter共laser inten-sity, focus position兲, emphasizing how phase matching can favor the one or the other contribution. Phase matching therefore serves to the “up-stream” control of the path selec-tion. A simple method of path selection after the generation process or “down-stream” consists in filtering the harmonic emission either in the spatial or in the spectral domain. Short or long quantum path can be, respectively, chosen by adjust-ing in the far field a diaphragm共on axis emission selected兲 or a disk共off axis emission selected兲. This quantum path selec-tion and optimizaselec-tion should lead to the generaselec-tion of regular and intense attosecond pulses.

ACKNOWLEDGMENTS

This research was supported by the Marie Curie Research Training Network XTRA 共Grant No. MRTN-CT-2003-505138兲, the Integrated Initiative of Infrastructure LASERLAB-EUROPE 共Grant No. RII3-CT-2003-506350, FOSCIL兲 and the New Emerging Science and Technology Contract No. TUIXS共NEST-012843兲.

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MERDJI et al. PHYSICAL REVIEW A 74, 043804共2006兲

Figure

FIG. 2. 共 Color online 兲 Average visibility of the fringes in the far-field interference pattern for the 17th harmonic, generated in argon, as a function of delay 共 1st-order autocorrelation trace 兲 : short 共 full circle 兲 and long 共 open triangle 兲 trajec
FIG. 3. 共 a 兲 Spatial profile of the 17th harmonic generated in argon with a laser energy of 0.85 mJ when the laser is focused 4 mm after the gas jet
FIG. 6. Measured spectral 共 a 兲 and spatial 共 b 兲 profiles of the 17th harmonic as a function of the gas jet position relative to the focus.
FIG. 7. 共 a 兲 Contribution of the short trajectory for H17 as a function of the relative jet-to-focus position z at E = 1.25 mJ and laser diaphragm d= 12.5 mm 共 open circle 兲
+2

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